The function \(f(x)=\begin{cases}\displaystyle\frac{|x|}{x}, & \text{if } x \ne 0\\0, & \text{if } x=0\end{cases}\) is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Discontinuous at x = 0
Since |x| = x for x > 0 and |x| = −x for x < 0, the function is
\( f(x) = \begin{cases} -1, & x \lt 0\\ 0, & x = 0\\ 1, & x \gt 0 \end{cases} \)
At x = 0 the left-hand and right-hand limits are
\( \displaystyle \lim_{x \to 0^-} f(x) = -1 \)
\( \displaystyle \lim_{x \to 0^+} f(x) = 1 \)
They are not equal, so the limit at 0 does not exist and f is discontinuous at x = 0.
At any other point f is constant (−1 or 1) near that point, so f is continuous at every x ≠ 0. Hence it is neither continuous for all x nor discontinuous for all x.
NCERT (2026-27 edition):
Continuity
